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Masters Theses Student Theses and Dissertations

1970

The evaluation of radar guided missile performance using various The evaluation of radar guided missile performance using various

target glint models target glint models

Franklin Delano Hockett

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Recommended Citation Recommended Citation Hockett, Franklin Delano, "The evaluation of radar guided missile performance using various target glint models" (1970). Masters Theses. 3584. https://scholarsmine.mst.edu/masters_theses/3584

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THE EVALUATION OF RADAR GUIDED

MISSILE PERFORMANCE USING VARIOUS

TARGET GLINT MODELS

BY

FRANKLIN DELANO HOCKETT, 1933

A

THESIS

submitted to the faculty of

THE UNIVERSITY OF MISSOURI - ROLLA

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

Rolla, Missouri

1970

Approved by

(advisor)~~~~~~LL~~~

79 pages c.l

ABSTRACT

A radar guided missile seeker approaching a target

arrives at a crossover point beyond which the entire

target is illuminated by the missile seeker's radar

antenna beam. When this occurs, for a complex target,

the radar senses a time changing target center location

which may greatly complicate the terminal tracking por­

tion of the seeker's flight. This thesis documents

various glint models in use today and compares the per­

formance of these models by using a seeker model to

determine the effect of glint on terminal tracking

performance.

A large volume of data has been compiled describing

various radar characteristics of complex targets. Some

of the glint models discussed herein are the result of

independent investigations and some models are derived

analytically. The return energy transmitted from a radar

and reflected from a complex target is generally statis­

tical in nature due to the random nature of the reflect­

ing surfaces dispersed over the target vehicle and also

because the target is continually changing aspect.

A radar target's reflecting characteristics are of

concern when a seeker is attempting to acquire the target

and again when the seeker is closing on the target. A

ii

large amount of data exists for determining long range

acquisition capabilities of various radar schemes.

The acquisition of a target in a sea clutter back­

ground has been thoroughly investigated in the past

decade and today's improved techniques (Moving Target

Indication and Pulse Compression) have enabled moderately

powered radars to acquire very small targets in high sea

states. At long ranges, the individual target elements

present a unified amplitude response (or appear as a

point source). The main problem during acquisition is

to reduce the viewing area to dimensions comparable to

the target so that the target return will be distinguish­

able. This argument does not apply to an interferometer

as target phase variations will still contribute to

acquisition error, however the sophistication required

to implement interferometers into a seeker design pre­

cludes their use for present day tracking schemes.

Although acquisition problems can severely limit

the response time of a seeker bearing vehicle or aircraft

to an eminent attack threat, the inability of a radar

seeker to operate in the presence of angle glint can

render the seeker useless. A seeker's design must often

be a compromise between tracking accuracy, acquisition

capability, and dynamic versatility, cince the weight

and cost penalties associated with multiple radar

tracking modes within a single seeker are prohibitive.

iii

Therefore, since the literature abounds with

acquisition theory and techniques, the main portion of

this thesis consists of analyzing the post-acquisition

seeker performance in the presence of angular glint or

angle noise.

The glint models used in this analysis represent

models in use at the present time. In addition, models

which were derived are compared to determine the degree

of correlation which exists between measured target models

and those models which are derived from their statistical

characteristics. The method employed for the evaluation

is general in nature so that the procedure could be used

to evaluate any subsequent glint model derivations. In

addition, some recent work is summarized which demon­

strates the advantages of using swept frequency techniques

to improve radar tracking performance in the presence of

angular glint.

lV

TABLE OF CONTENTS

PAGE

ABSTRACT . • . • . . . . . . . . . . . . . . . . . ii

LIST OF FIGURES. . . . . . . . . . . . . . . . vii

LIST OF SYMBOLS. . • . • . . . . . . . . . . . viii

I. INTRODUCTION . . . 1

II. SEEKER DESCRIPTION . . . . . . . . . . 6

A. GENERAL . . . . . . . . . . . . 6

B. SEEKER FUNCTIONAL DESCRIPTION . 7

c. DERIVATION OF ERROR SOURCES . . . . 13

III. TARGET DESCRIPTION . . . . . . . . . . 17

A. GENERAL . . . . . . . . . . . . . . 17

B. TARGET OPTICS . . • . • . . . . . . 18

c. TARGET STATISTICS . . . . . . . . . 29

IV. GLINT MODEL EVALUATION. • . . . . . 40

A. GENERAL . . . . . . . . . . 40

B. SEEKER PERFORMANCE AS A FUNCTION OF GLINT. . . . . . . . . . . . 40

v. A RECENT GLINT REDUCTION TECHNIQUE 45

A. GENERAL . . . . . . . . . . . . . . 45

B. ANALYTIC IMPROVEMENT FACTOR . . . . 45

VI. CONCLUSIONS . . . . . . . . . . . . . . 54

vi

Page

VII. Appendices

A. Proportional Navigation . . . . . . 56

B. Far Field Criterion for Plane Wave. 60

c. Angular Error For the Two-Element Target . . . . . . . . 63

VIII.Bibliography . . . . . . . . . . . . . 68

IX. Vita . . . . . . . . . . 69

FIGURE

II-1

II-2

II-3

II-4

III-1

III-2

III-3

III-4

III-5

III-6

IV-1

V-1

V-2

V-3

AI-l

AI-2

AI-3

AII-1

AIII-1

AIII-2

FIGURES

TITLE

General Missile Guidance System .

Proportional Navigation Scheme •.

Simplified Block Diagram for a Disturbance Input . . . Seeker Simplified Block Diagram •

Tracking Radar Lobe Pattern . • . . . . Tracking Error For Two-Element Target •

General Missile Coordinates

Space Geometry . . . . . . . . . . . . Glint Spectral Density Example.

Probability Density Function for Sum of Two Normal Distributed Variables

Glint Produced Rate Error .

Histogram

Frequency Agility Glint Improvement •.

Glint Autocorrelation Approximation . .

Proportional Navigation Geometry .••.

Polar Coordinates •

Evaluation of Polar Coordinate Unit Vectors. • . •..•.

Antenna Pattern Geometry

Radar Angle Error Coordinates •

Wavefront Geometry •.••...

PAGE

10

13

15

18

20

24

25

36

39

43

49

52

53

56

56

57

60

63

65

vii

List of Symbols

Sj,v - IF Signal-to-Noise Ratio

~ - Transmitted Power

~ - Antenna Gain

A - Wavelength

R - Radar Range

k - Boltzmann's Constant

~ - Constant (general)

Cf - Radar Cross Section

T - Absolute Temperature

8 - Receiver Noise Bandwidth . ~ - Missile Velocity Vector

A~.s - Line-of-Sight Angle . AJ..s - Line-of-Sight Rate

ArL - Antenna Angle

G, - Radar Shaping Network

Gz Radar Amplifier, power amplifier, and actuator

antenna Gimbal

~ - Rate Command

~ - Disturbance Input

KG - Gain of the Gyro Feedback Path

viij

s 5-/ LaPlace Transform Variable

LU - Angular Frequency Variable

E r - Tracking Error

l<p - Position error coefficient

Kv - Velocity error coefficient

4 - Acceleration error coefficient

An - Time Derivative of antenna angle

e l

- Time Derivative of input disturbance

e., - Radar antenna crossover point on receiver antenna

base line

~~ - Relative phase difference for the returned signals

from multi-reflector targets

~ - Ratio of target amplitudes

e#- Individual reflector's (target) amplitude

B~ - Target Angular coordinates

~ - Incident power

~ - Electric field strength vector

H - Magnetic field strength vector

/Z - Intrinsic impedance

d~ - Distance of elemental target reflectors from

receiver

E th - The reflected field produced by the k target r;:,

element

j ';{

-n - Number of independent target elements

~ Half-power frequency (noise bandwidth)

UJa, - Rate of change of aspect angle

h - Maximum dimension of the target

¢fo) - Zero frequency spectral density

~· - Standard deviation of a statistical distribution

11(,.· - Mean of a statistical distribution

~ - Loss Factor

/V~ - Receiver noise figure

X

I. INTRODUCTION

During the past decade, a considerable amount of

effort has been expended in perfecting existing radar

techniques. The radar industry has been saturated with

new circuit and system techniques. The art of control

dynamics has been perfected to the extent that an

enormous volume of material exists describing the appli­

cation of these methods to various types of controlled

vehicles. New and more sophisticated transmitting and

receiving devices have been introduced into the radar

industry. The radar industry has reached a high level

of maturity in developing radar equipment of all descrip­

tion and for many various applications.

The largest problem which confronts the radar system

designed is that of being able to adequately describe

the radar's operating environment and its effect on

radar performance. In addition, the radar designer must

be capable of deriving radar cross sections which will be

an adequate basis for radar design.

For the present discussion, the radar application

is that where a radar is used to assist a missile in

acquiring a target and then tracking the target to the

point where the missile proximity fuse will dispose of

the target (for the purposes of discussion in this article

it will be assumed that the missile will impact with the

l

target - although this is not a necessary condition for

successful missile performance).

Two primary areas of interest for an air launched

anti-ship missile are the background clutter from which

a ship (or target in general) must be detected (a long

range effect) and the disturbing influence of angular

glint on the seeker's tracking characteristics (a short

range effect). A ship will be used for the analysis of

this paper since the problems associated with a moving

target are best illustrated by a conservative target es­

timate. A more dynamic situation would be the inter­

ception of an aircraft with a missile seeker, however,

the increased analytic complexity does not seem

warranted since the basic technique, not all of the

potential applications, is the intended goal of the paper.

The problem of clutter has received a great deal of

attention in the literature. Nevertheless, acquisition

represents a very important phase of a modern anti-ship

weapon system since the response time of an attacking

aircraft equipped with missile seekers is related to the

ability of the seeker to lock on the target. In

addition, the more favorable logistics of increased range

acquisition handicaps a potential enemy in detecting

and locating the carrier aircraft.

2

Although less frequently discussed, the problems

associated with angular glint can be much more damaging

and generally increase the design difficulties. At

large ranges, assuming acquisition has taken place, the

target can be assumed to be a point source and the well

known radar equation can be used. For non-cooperative

tracking (using reflected energy) the relationship for

the IF (Intermediate Frequency) signal-to-noise is given

by1 as:

where S/# is the receiver signal-to-noise ratio prior to envelope detection but after IF amplification

~ is the transmitted power in watts

G is the antenna gain (this assumes the same antenna is used for receiving and transmitting)

R

)(

T

B

is the radar wavelength in consistent units

is the radar cross section in consistent units

is the Radar Range in consistent units

is Boltzmann's constant 1.38 x 10-23 watts per hertz per degree kelvin

is the absolute temperature of the signal source in degrees kelvin

is the equivalent noise bandwidth of the receiver in hertz

~ Loss factor, numeric

/Y~ Receiver Noise figure, numeric

The radar cross section term,cr, is the most

difficult term in the radar equation to describe

analytically. For relatively simple geometries, analytic

solutions can be obtained and these are listed in

reference 1. For more complicated targets either a

physical model must be constructed and evaluated using

an antenna range or a mathematical model must be derived.

In either case, some means is required to determine the

effect of the target cross section variation on radar

performance.

Now assume the target is being tracked. A thorough

discussion of acquisition techniques can be found in

either reference 1 or 2. As the range to the target

decreases glint becomes more pronounced. The critical

point of the missile flight path (with respect to glint)

occurs when the target subtends the tracking antenna

beamwidth and small variations in pointing commands at

this range represent large disturbances to the missile

(vehicles) control dynamics. A definition of angular

1 . . b 1 g int lS glven y as: "The disturbance in apparent

angle of arrival due to interference phenomena between

reflecting elements of the target." Therefore, the

degree of success in designing a missile seeker depends

to a large degree on the systems engineer's ability to

determine the glint spectrum for the anticipated target

vehicles. Here again, where exact surface features are

4

known (a condition not likely to exist for non­

cooperative targets) a physical model will provide a

good basis for determining the glint spectrum. Where

this data is missing, one has to either use similar

target characteristics or provide a mathematical model

sufficiently similar in large dimensions to provide

the required data. For a target with any complexity

the analytic method must use a statistical approach.

The final step in this analysis is to provide a

comparative basis from which design trade-offs can be

made. To accomplish this a general seeker model will

be used, and since the intent of this paper is not to

design a missile seeker, a conventional seeker scheme

will be used for comparison purposes. During the last

three years frequency agile radars (swept frequency

radars) have been used to increase the acquisition

range and reduce the glint spectrum effect on tracking

accuracy. A discussion of this new area of research

is included.

5

II. SEEKER DESCRIPTION

A. General

Since the primary objective of this analysis is to

derive a glint model, the radar model and control dynamics

scheme were selected from an existing design described

in reference 2 This was done to prevent including a

large amount of analytic detail which is present in a

great number of periodicals and standard radar textbooks.

Also, the final conclusions concerning the developed

glint model can best be determined by comparisons made

with existing models using a familar radar tracking

scheme. The tracking scheme used is contained in chapter

9 of reference 2. Basically, the tracker design used

will be a proportional navigation scheme. The dynamic

equations for this seeker are developed in appendix I.*

The radar seeker steady-state error as influenced

by the tracking loop bandwidth is analyzed for a point

in the missile trajectory close to the target. The means

by which the seeker arrived at this point (acquisition

and inertial guidance) will not be dwelt on as this is

another phase in the seeker designs where the effects of

*A brief description and the equations of motion are contained in appendix I.

6

receiver noise, clutter and general transmission

quantities must be evaluated. The ultimate performance

criterion for a radar seeker design in the application

of a guided missile is the missile miss distance.

However, the sensitivity of this parameter to various

error sources requires that we consider the design of an

entire radar seeker-autopilot-airframe flight dynamics

loop, which is beyond the scope of this investigation.

The discussion, therefore, is restricted to the analysis

of errors in the rate commands generated by the seeker

(which are supplied as an input to an autopilot in a

proportional navigation scheme) as a function of

line-of-sight variations.

B. Seeker Functional Description

Figure l is a diagram of the missile control system

for one channel (azimuth or elevation) of a proportional

navigation scheme. In such a scheme, the angular velocity

• • of the missile's velocity vector,o , is commanded pro-

portional to the missile target line-of-sight rate; . • A. ., • 1. e. K"= 1' t.s where q is the constant of proportionality

and ALS is the line-of-sight rate. The seeker descrip­

tion of figure l encompasses the tracking and

*Characters dotted represent derivatives with respect to the appropriate independent variable.

**See equation AI-9, Appendix l.

7

.DISH ANGLE /N S.l'~ce"

~""'OOI"lc

TARGG"T 4N6LG /AI S.PACIT

TA.RGtFT

AP.P-4.RENT TA/?$ET -4-NGUT

/N ~-4CE K I /"'70TION

<?ai'?PuTe'~ ANGla P?AcK S/~NAL

STAOILIZATldA/

S/~N~L.

I I I I I /(INEr?AriC

I FtE08-4CK

I I I I I I I

A/~AA-ru: I 1 FL/tli#T P,qrl-t ANtSu:z· 1 I >'! AND

Aul'lJP/~ar /"1/SSILE" • TA/tGt:T

FIGUEE II-1 GENERAL MISSILE GUIDANCE SYSTEM

0+

stabilization loops. Generally speaking, glint produces

the largest effect in the azimuth channel where the

target aspect can be in the neighborhood of an order of

magnitude greater than the elevation.

Now the radar seeker design must allow the seeker's

antenna to remain pointed at the target during target

or platform motion, however, the response must be slow

enough to prohibit the seeker from responding to noise

inputs to the radar receiver. Therefore, the seeker

design represents a compromise whereby the loop response

needed to reduce antenna platform motion is obtained

with the stabilization loop, and the slower tracking

loop is designed to provide noise filtering without

causing large dynamic bias error. Figure (2) is a

diagram giving simplified transfer functions of the

variable components. The stabilization loop includes

the antenna gimbal, gimbal actuator, and actuator drive

amplifier, a rate gyro and shaping networks. Examination

of the transfer function ArL/e indicate~c; that a high

loop gain is necessary to provide stabilization.

For this situation the transfer function~ i~-; V;z

approximately '/J<6 S where k6 is the gain of the gyro

feedback path. The tracking loop includes the radar

receiver, the angle tracking demodulation, amplifier,

bandvvidth shaping circuits, and the stabilization loop

9

74Rt:Gr l"''d716N

A'A0-4~

1" :.ItA CKIN6 Ld6P

VA

4CTUAr()R Gl N i3<4 L.

.3 TA8ti.I~A 7'.14A/ ~tJdP

l RAre GY~o

e

.----------~8

• I Aorc.P~t.or I r

~ AIRPifANE 1---=-

SP.ItC.I¥ A~NG;o?Aru::#

FIGURE II-2 PROPORTIONAL NAVIGATION SCHEME

TYPE I STABILIZATION LOOP

f--J 0

already described.

A typical Transfer characteristic of the tracking

loop is:

Where,

and

q, = A; (I -1- 5/u.JE) (/ -1- 5/~)(11- S/w.:J)

This gives

:: ATL == Er

The frequency where the amplitude of Gr is zero

Db defines the open loop bandwidth, We , of the tracking

loop (often referred to as the crossover frequency).

Typical break frequencies for this type of servo give a

slope at crossover of 20 Db per decade providing

stabilization of the tracking loop. •

Random errors in the tracking rate ALs are made up

of gyro inaccuracies, radar noise transmitted through

the tracking loop to the gyro output, and antenna rates

caused by the platform motion which is attenuated by the

~;tabilization loop. Bi.::J::c) errors are contributed by the

~~ervo ~;Lc.::1dy-~;tatc; and transient errors and are a

funcLion of the: E3ervo type and servo bandwidth.

11

12

A small discussion is included on the error sources

to add continuity to the discussion, however, the

majority of this article is occupied by the treatment

of the glint problem. In addition, the error dependence

will be shown. This is done since for the majority of

situations, the bandwidth of the system is the means by

which the servo engineer accomplishes the compromise

between system dynamic response and noise sensitivities.

Figure II-4 contains a table listing the principal

error sources in addition to glint which is considered

for this analysis to be the major component of radar

noise at short range. The expressions for the errors

are developed as a supplement to the table. For this

radar seeker model, gyro bias error is con dered

negligible.

To evaluate the error coefficients, one

expands I 1 t- Gt's)

into a Maclaurin series into the

form, I I

/ t- Gt's) I -1- l<p

These coefficients have been evaluated and are listed

in reference J and the values are:

kv = )<

J<A = I< J< (1 /w, t-- 1/~ -I /wz) -I

e

Some simplification can be made if the following

assumptions are valid,

GU,/Wz <-< ~ 4./3 >> ~_, u:.tz

,.... ~ ,~<w, ( .5/we ,to/) I..;Jr- 52

and for this function

and, k v 1 (~-, , ) V :: "A 1tU1 : We c.v 2 ,w,

C. Derivation of Error Sources

Stabilization - The following figure represents

the seeker simplified block diagram redrawn for a

disturbance input e.

~------------------------------------------~47L

FIGURE II-3

Simplified Block Diagram for a Disturbance Input e

13

The transfer function from output to input is:

-4rL e

I / r Gz (6, -J- G.3 )

I

The output rate error for a disturbance rate & is:

.

This results by again reducing the blo~k diagram and solving for Art. as a function of the input e

• Now if 9 is a pure sine wave of amplitude cQ, , the rms

error in An. is: . ArL == 0. 707Bo

/G.e G3 /

14

A~s

.f?AJJA,q NOIS~-

Er

4~s: L..tNG"· t:JF- S/~N-r ANISC:.G"

A7L = ANTE"NNA 4N~G6"

~ = RAre coi"''I"''ANO

~

k, (.Ywz'f-1} (.S/w, .,., ){SfwJ +I)

6/ ..., ,/?AJ>,IIR S#AP/NG Ne7\AIDRJ.(

Gz = SH~PIAIG /Ve//Ait!JRI< {.P~W(;""/t AHPL.J..C/~

Ac.7vAro~/ erG.) FIGURE II-4

G2-

X a.

/'?ISS/t.V .1'-?crn>N

S(,....s +.t) I • ~4rL.

G3

KaS

~ /NG"-ac-- Sl't&l'ftlr ~ATe ra 4urc,o/£cr

SEEKER SIMPLIFIED BLOCK DIAGRAM

'--

ERROR SOURCES

RANDOM

GYRO RANDOM

ANTENNA MOTION

BIAS

THACK LOOP

TABLE II-1 TABLE OF LINE-OF-SIGHT RATE ERRORS

APPROXIMATE RELATIONSHIP

• 8 ALs

• 0. 7£?7eo

Gz.G3

• • • •• At.s A,s -- --l<v l<a..

REMARKS

o. OOlL-8 L. tJ. 02

Single Frequency Input Rate. Peak Amplitude

For

Gr = I< { 5/wz .,.; ) S(Sjw,-rl )(SJ~ -;-1)

III. TARGET DESCRIPTION

A. General

In describing the target, the glint problem must be

defined in terms of the target size and the beamwidth of

the antenna to determine the range at which the glint

noise spectrum becomes significant. Two methods are used

to determine the effects of target glint on terminal

tracking behavior. The first is an analytic approach

fashioned after the analysis of a two element target

described in reference 1. The second approach is to

describe the target statistically and use the results

of this development as the glint noise spectrum input

for the seeker. For the statistical approach, the target

will be assumed to consist of a collection of individual

targets whose scattering properties can be described

statistically. Also, a known glint spectrum is used

for comparison purposes.

Now, one would expect that the majority of these

scatterers would operate in the surface and edge

scattering region. The reason for this is that for good

acquisition range (high antenna gains), nominal weather

performance, and present state-of-the-art design

capabilities most anti-ship radar seekers are presently

concentrated in the region above 10 gigahertz. The

smaller antenna also provides a larger target-to-clutter

1'i

discrimination although the smaller beamwidth compounds

the dynamics of acquisition. This region of surface

scattering is one where the optics case is being

approached, therefore the geometry techniques employed

in optics can be applied to obtain approximations to the

scattering problem.

B. Target Optics

An analysis of an n-element target can be made, the

analysis being similar to that of Locke in reference (l)

page 440.

For this analysis the function for each lobe of a

tracking radar antenna pattern may be expressed as a

Taylor series about the crossover point as seen

in Figure III-1.

r '/ ) "" 2 flB) =--r~) t- f' £B-Bo .;._!_I?&-~).;----.?/

A s

r 2P8

Figure III-1, Tracking Radar Lobe Pattern

18

For small angles off-axis, only the first two terms

may be retained, which is true for an antenna response

which is a linear function of the error angle, and this

is the case for targets which are close to the boresight

axis. Then the amplitude response can be written as,

(3-2)

and for a single element target, the signal from lobe A

is

(3-3)

and from lobe B is

e 8 = G[l f-p(&;r-~")] CO.$wt: (3-4)

where G is defined in reference 1 as lumping together

target size, system gain, etc •• The detector squares

eA andes and a low-pass filter rejects all but the d-e

term and low frequency terms and since appropriate

filtering is used to filter out the second harmonic,

t:- = G~ (S>r- cO~)

and for an on-axis target, cb =Brande = o.

Now if the target consists of two reflecting elements,

each will contribute to the received signal. The returns

will be at the same frequency but any phase depending upon

the relative range. The nature of the problem becomes

evident at this point since at relatively long ranges,

where the target span is less than 5% of the radar

beamwidth the target will appear to be a point source and

19

will continue until the radar antenna is capable of

resolving the individual target elements, i.e. at

relatively long ranges the return energy appears to be

concentrated at a point source. Reference 1 containE

the solution for two targets with the results included

in graphic form. The results are included in this paper

for reference purposes in the form of Figure III-2 •

0

. I

I<= ~:1

I I

I

.1(= .RAr.ttJ t>F "'""'urv.cl' I 41(: ~ :rM6t!T / ret r-'IR~I!T 2 I o . .,

I I

I I

:/ t."': -I t1.~ ,, .-

~,,....... )..t~ '/ I tJ, J

"'I-~ .~-, /.,. .,. ..,. I

A"'J-~'-- ,.. L - - - ;; .::. ~.... ,, .k:

-- ... - -; -:...-.: .=--=..: -:.. -=--------:.. ~, ""1 0. z _-_:..;-~-:-_-,:-----_-_ --":. ==_..--*"I - ,

- - - _,_ -- - - =• - - - - ..:~- - - - - I J~· ~o0 9o 0 /20° /~o·

/iFLAT.IVG P#ASE AIVGtE

Figure III-2, Tracking Error For The Two-Element Target

Locke's result is,

e<' = B7; r G>.o 'lz +-a.. C!~5-rr ( 3-5) / ;1- q z + Pa. C4S _.,.

where, oc- is the relative phase difference for the two

target ret urns

~r/ is the off-angle for target 1

and eo is the angular spacing between the target

elements.

Now equation 3-5 is plotted for varying phase

difference (OC). Two cases are of interest, first

for ~ = o, the two return signals are in phase. Then

the coefficient of eo in (3-5) will reduce to a./(q_?'-1)

and for 180 degrees the coefficient reduces to ut)f~-1)

For the second case the angle of zero error will be out-

side the limits of the two target elements.

Now for n = 3, ea = G[ 11- p (B.r; -~)coswt: + a,6[1o~-p(Srz-f!{,j}os(cu"C10f)

-1- azG[I+,o(&;;-t:b)}cos(eut: +D<3) (3-6)

and,

eA= G[;- p (Br,-Ba)}co~wt'o~- a,G{!-p (erz-4)]co.s(wtroc;) ( 3_7) + Qz G[/-p(er-3 -B<J)]cosf4.1t--;-crL)

where ~, is the amplitude ratio of target element two

to target element one, and Qz is the amplitude ratio of

target element three to target element one, and o<, and

~~are the phase differences between target element one

and target elements two and three res~ectively. Now to

proceed with the development, eA and ea must be squared.

The simplest approach appears to be to make the follow-

ing substitutions, For eA

A= G[t-p(Br;-l:l~)'J

8 = a.,G[!- p(BTe-Bo)j

C= ClzG£1- p(Bra-~)J

(J-8)

(3-9)

(3-10)

21

Then,

(3-11)

and expanding this and using the fact that appropriate

filtering will leave only the low frequency and DC term, :i .2. ZJ

eA 2= A/z f' B/,e '~- C /Z f- AB Cos-; r AC ca.s ~z.. ~8C(Ct!Ui*; CdS"""Z.-;- :Sn111!1tj s.~ncr~)

Likewise, for ea

then,

.4 "= G [ / o~- p (t97j- B"').J

B "= t:z,. G [ / ~ ,Q ( lh;. - ~It> ) J e ': Clz G[; rP (BJj- £;")]

/~ ,.c ,& ; / , / C?tJ~ Aa re/& ,C'/2 rAA3 (!c:so.;.;./Jc. ecs~z.

~Be"(' (!as .::v; ac s ~ ~ sn,.:v; s ... n ~)

(3-12)

CJ-13)

CJ-14)

(3-15)

(3-16)

substituting the corresponding values for A, ~ C!, A,"

8 ', and C' into the expression G = e 8 ~ e A z. £-= //z{G r.-9 p{t!Jr;-6io} r Q./G ~~p('&:lr~- ~) .,. ~ 2 G z.-"'p (B7j -~J}

-r-r:Jccs~ ~ .. t; ~[(Sr,-~.,) r {&r, -4)};~- ~~c.s~z 4zGp{B7j-4;)+(~-4}j

(3-17)

For the on target conditions G = CJ and

~ = ~,_c?, ~7.! rt?L~l3 rt?.cc.:s'"1{c9r, '1-~ ~ao~~"~Jrll,qj"a.t-r~,..s-:s•.,-;)( 3_H3) [t-~oq, .. ~t~z ... ,..~,f!c~-t; +Zttzco.s«;. t-:lQ, t?~(eos~ Co6otl ~.sn,-: snrD', )1 (1!97£.,.. t:!cr)

Now for purposes of extracting some useful information,

assume

!9JZ.. = {;) T, ,. ~ i:J

and

') ') ,,.(

Therefore,

B. = GJr. 1- 6Jb[ Cf, ~ Qz. 2+- Q, Co~ ...... , - Qz. co:. oro"- :J 0 , z ( )

/rtl, +tl,.z-~o2Q,Cosar, +-..?<7zCos~ r..:7Q,~(Q,.so;;c-0s~ .Jsu1...-;su1a;;_) 3-19

To evaluate equation (3-18) for all possible conditions

would require the use of a computer and a considerable

amount of programming and operating time. This does

not seem justifiable when the target in reality is

statistical in nature and one would penalize the seeker

design by attempting to apply equation (3-18) or its

successor for a model employing more target elements.

However, a few cases will be investigated by using

equation (3-19) in an attempt to arrive at some general

considerations. For the first case, assume a, is equal

to az and the respective phase differences are zero.

From equation (3-19), the indicated error is zero. For

case two, assume a/ is equal to a~ and C><', = o and

, which indicates

a fixed error. Now ifo<;= D<z. = 180°, then 9 0 = 9r +-

e 0 (o) , and again there is no error. The error will

become infinite at those points where the denominator

of (3-19) approaches zero, rearranging,

Ql. +- a.{cc~-"; .,.. C.d .s or~-) ~ I :6 (3-20)

I-I" Ct:J:J-. Cc :SOil. .,_ SU'tOI'; S.1~<V~ 2 (I +co.:s~ co.s orz.. +smor, sm...-.~.)

Now this is a quadratic in a that could be programmed

on a computer for solutions, if any exist. The apparent

difficulty in finding a root to satisfy (3-20) might

indicate that as additional target elements are added

the severity of the tracking error is decreased. The

solution of (3-20) and the number of roots obtained

would either confirm or deny this assumption.

Figure III-3 illustrates the general missile coordinates

which will be used for purposes of discussion.

--

,('

o<":AN'GtE oF ATTACK

C: //II{ GET I!JiR4R AN~

At.s::£,-Aie'-"~·.JI6NT AN~l.t:

~: 4/TtT'-'Dff' ,1/MS"-8

------r rY::AII/t;te oF" /He" .c-a,NT

Figure III-3, General Missile Coordinates

24

Also, the required space geometry of the seeker and

target is illustrated in Figure III-4.

Figure III-4, Space Geometry

The angular velocity of the line-of-sight in space can

be derived by finding the normal components of the

missile velocityV,~ and target velocity Vr with respect

to the line-of-sight. From Figure III-4, the expression

is, •

- Ats = v/'7 $;/\/ rr- 4t.s) + Vr SIN a (J-21) ~ R

where,

VM = missile velocity

Vr = target velocity

R = The range between the missile and the target

At..s = Angle of line-of-sight in space

~ = Angle of the flight path vector in space

.J!3 = Angle between the target velocity vecto:~ ar:d

the missile-target line-of-sight.

. l I ' '

Now the tracking geometry shows a dynamic environment

where the relative motion of both the missile and the

target provide kinematic feedback to the seeker tracking

loop (see figure II-1). The error angle derived from

the seeker antenna is directly related in both magnitude

and sense to the error in heading of the missile.

Now for a missile airframe perturbation an error

voltage is developed at the receiver output terminals

which may be expressed mathematically as

E = J<., ( T- 4t.s) (3-22)

and, a rate of change of the angle/ occurs which is

proportional to the input displacementi"- 4,t.s. This

may be expressed as •

(3-23)

The parameter W, is equal to the ratio of the missile

velocity divided by the range between the missile and

the target multiplied by a proportionality constantf<z.

Therefore, the combined error becomes the sum of two

term~). Letting 1<., equal Kz. and integrating both

sides of the expression for the rate signal,

(3-24)

where

sS J

- The Laplace Transform variable

~ Assumed constant.

26

When the missile is at long range~. is small since R

is large, and UJ1 is small with respect to l and the

effect of the integration term is small. As the range

closes, the corresponding lag caused by this integration

can cause instability (see reference 1 , page 628).

The remaining term of equation 3-21 is considered

next. In contrast to the preceding discussion the

second term of the equation deals with target motion.

The target motion is reflected as an error at the seeker

antenna and~ can be expressed as, c

/:. = V7 j Suv B dt: 7 D R_

and this can be expressed as

t/8 = \/rj t S/NB dt' B (:1 A

If the assumption is made that during an interval of

time the angle)1 and the range have particular values

and are not functions of time, and since!;- may be

taken to be constant lr/13 takes the form

h/a - tuz/s (3-25)

where wz = Vr S/N.B

1<1!3

The value of uJLvaries with target velocity, the angles

of the space geometry and the range to the target. Again

at long ranges, the value of~~ is small so that target

motion has very little to do with the rate of change of

the line-of-sight angle in space. However, when the

27

range decreases then small perturbations in the target

or missile motion may cause large variations of the

target angle.

The preceding discussion illustrates that at short

ranges where bandwidths are of necessity required to be

larger to reduce error (Larger bandwidth here inferring

shorter time constants) the rms value of the glint

spectrum becomes larger, as more of this noise is intro­

duced due to the increased bandwidth. Therefore, the

next subject for discussion is the development of the

glint spectrum.

28

C. Target Statistics

Basically radar targets behave as parasitic antennas

and can be grouped into two general classes. The first

class has essentially fixed geometry such as a cylinder

or sphere. The second class has a geometry which is

best described statistically.

The radar cross-section is a transfer function that

allows us to go from the power density in the plane wave

incident on the target to the power delivered per

unit of solid angle in the direction of the receiving

antenna. (All discussions in this article assume a far

field antenna pattern, i.e. an incident plane wave at

the target). The plane wave criterion is generally z

considered to be in effect for ranges ( R) ~ ( 2 D) where ).

D is the diameter of the largest aperture. (A discussion

is included on this subject in Appendix II). Now the

scattering cross section can be analyzed using the

following approach. The power collected by the

r·ccciv-Lng antenna i~-, [The power per unit area at the

t:u·gctJ X [The power delivered by the target per unit

solid angle in the direction of the receiving antenna

per unit power density at the target] X [The solid

angle of the effective collecting aperature of the

radar receiving antenna, as viewed from the target].

29

The solid angle of the receiving antenna is, by

definition of solid angle,Ae/Rz Therefore,

[Power delivered per unit J solid angle in the direction Ae of the receiver per unit --z power density at the target ~r

(3-26)

Now we replace the real target with one which reflects

isotropically but still delivers the same power per unit

solid angle in the direction of the receiving antenna

(per unit power density incident at the target), then the

total power that this isotropic reflector will delivery

will be 41f times the middle term in equation (3-26);

this power will be uniformly distributed over a sphere

at any distance from the target. The fraction of this

sphere occupied by the receiving antenna is ~ _1_ • 'R;- z "'iT

Therefore, if we define the target's radar cross section

of the isotropic reflector which delivers the same power

per unit solid angle, per unit incident power density

in the direction of the receiving antenna, as the actual

target does the radar equation becomes

s== I Pt;Gr) {u)( Ae) (_L) (_.;;,. Rt '- Rr ~ -? .l/ /

(3-27)

The previous definitions can be combined to express

the received energy as: the power received over the

solid angle of the receiving antenna aperature is equal

to [The power incident at the target per unit area] X

~X -L- [The solid angle at the receiving aperture]. 11f

30

Or,

CT ::: "'i'JT ln the dl re ctlon of the recel ver power per unit area incident at the ~~ower de~iver~d per. unit sol~d angle]

target ( -28)

The power delivered to the surface area of a unit solid

angle, at the receiver, is P .... R.,..z. where P,... is the power·

per unit area in the plane wave, at a distance Rr from

the scatterer (i.e. at the receiving antenna). If P..:.

is the power per unit area incident at the scattcr·er,

then,

cr=

Now the average power density of a plane wave is

P= j, /£/J/1/= j, ~~~; -n./#/ 2 (J-30) -

where E = Electric Field Strength

# = Magnetic Field Intensity

"?( = The intrinsic impedance of the propagating

medium combining equations 3-29 and 3-30,

(J":: .-)IJT Rr :z. ( #r / 2.= -¥ 7T /?. z. / ~~ ,_

I .tl~j z. / E..i.-/ z.

and a far field definition of equation 3-31

(~-:1)

J. ~) '

o-: ~7/ L//"7 ~ zj Jlr;: --97T L1M 1?1' ¥-/ Er/ Z-

Ji)....,. -c:1 / ~.; z.. R;. ... ~ I ;.,;./ ~ This last expression is the one invariably used to

determine the radar cros::::1 section after the :::3catteri

problem for the reflected field :::>trength ha:::; been 'ol Vl'd.

·~ I

Now for large targets, where local surface currents

in one part are relatively independent of the current

in another part, the reflected field at the receiving

antenna is the vector sum of the fields which have

arisen from each component target separately.

If ~~ is the reflected field produced by the kth

element and dk is the distance of this element from

the receiver then, / ~ / ::: ~ t / gr~ I e ./ ( 11 lTd KIA- j (J-33)

Because, even at closing ranges, the parts of the target

are relatively close together compared to the distance

between the transmitter and the target, the incident

field strength E;· is approximately the same on all the

and substituting this result in equation (3-32)

- ~' -? \ ,-;;:;- ; { ~ 7T cl K.l)._l c. CT- L YO}< e

1::1

( -)-'S) J -· '

This equation will form the basis for constructing a

statistical target model. An example for a two element

Now using the previously developed equation, the

basic assumption required to justify the use of this

) ') ) I

expression is that the total field produced by a group

of individual scatters is a linear combination of the

fields that would arise from each individual element if

the elements acted by themselves. Therefore, the

elements act as independent scatters, and there is no

mutual coupling. In addition, the assumption is made

that the individual cross sections remain constant.

The time during which the glint spectrum applies

is short, therefore, the phase ~ll d~ is a random variable /L

which can take on any value from zero to 2h' . The

desired cross section, from equation 3-35 is the vector

sum of n vectors which have random relative phase.

Assuming that all of the scattering elements are

identical (all of the OJ:s are equal) and calling the

value of the cross section of any one element ~o ,

equation 3-35 becomes x .. ; ?tJTdx.../ e

cr= /L 'fer;, e ).. K.:./

(3-36)

Statistically this problem is identical to the isotropic

two-dimensional random walk where n successive steps

of fixed length O"o are taken and where the direction of

each step is completely random. If we consider the

)(-component of the kth step to be 'Vao cos ~;rd1::. and the rl

J -component of the kth step to be ~ sin -i~JTf:LJ(.. ' then

~

the joint probability density function of components

{ x; #) after ~ steps is,

This result can be justified as follows, for two sets

fksll)} and £kn (1:)] which are independent, the joint

probability density function is,

p (~ :nJ = J? Is);; (n) (3-45)

Now let,

j= S 1--n. in order to define y for a particular s, then

-n= ;;- s (3-46)

therefore for each s the integrand for the convolution

integral becomes

(3-47)

Now s can take any value from - ot::' to oe , therefore

summing over all possible s gives the result. Now let

'>{tjw) and V:z.LJw) be the Fourier Transform of P,~s) and

Pz/s) respectively, and these are defined by, ~

"'t tjw) = /- e -.;'ws fils)ds

>{ (}t..U) = J--" e -./UJ nJi.f:n) d.YL

Now from equation (III-47), .<tf/)

pl;t) = /_ ./?~s) 1?. (~- s) d s

and

therefore, or.:>

{otO -./.w(srn,) f /f/5)A tp-s)d:s ofj!. '/ {J'-v) = _J_ e -L ~

Equation III-53 can be expanded to show

(3-48)

(3-49)

( 3-50)

( 3-51)

37

Then, the probability density function ?f;.) l:'i g1 vt:n by

taking the inverse ~ransform of equation (3-5J)

or,

C3-52)

This last expression illustrates that the probability

density function for the sum of independent random

variables may be obtained by first determining the

Fourier Transforms of the individual distributions and

then by calculating the inverse Fourier Tran~form of

their products.

To illustrate the modified spectrum due to

additional target elements the individual target

statistical characteristics will be assumed to be

normal. Suppose that P,t.s) and 1;. IYL) are governed by

independent normal distribution such that 1 -{s-m)-z.l ,.._'Z... l1 ts) = - e / rf' v ~

a; Y.z.JT ( ~.'-[.)>) ' ..

and z.

1 - ( 77 - "J'Y?l. ) /~ o;_ 2.

/{ fn) ~ o;_ v.vr e Now to obtain ptfj) the Fourier Transforms of P,!5) and

/;_tn..) must be obtained.

The resulting inverse transform is

I - lp- 171) /~ tT z.. P tj ) :=: o- /i:ii- e

where

and

(3-57)

Thus the sum of two independent normals is also a normal

distribution* with the resulting mean and the variance

equal to the sum of the individual means and variances.

This analysis can be applied to the sum of many

independent variables. These results are illustrated

in figure III-6. ,PI'~)

I

-'· , I .. , I '

;'' r··, I '

I I

I

\ • \

/f.ls) 1 I

P.bt) , \

~I '*"r._,ll I

/

' ' f

I

'

I (

I I

I

I I I

I I I I

I

I

I I \

z_,;

\ I

' . ,

I

'

I

I I

16"u::"' f I

I '. I

\

\ I \ . \ ' \ . 1\

I \

I \

I \

J J

I

\ \ \ \

\

\

' \ ' ' , ... , / ' I

~----------~----~--~--~--~;~~-------'~------~1~----------------·~,--~x /YI, ""''- ~

Figure III-6, Probability Density Function for Sum of Two Normal

Distributed Variables

Therefore, the standard deviation of the total glint is

a function of the target size

*These are classical results, C.F. Reference (l) P· 390 ·

39

IV. GLINT MODEL EVALUATION

A. General

The previously defined model will now be used to

evaluate the effect of glint on the angle tracking

performance of this seeker.

B. Seeker Performance as a Function of Glint

The average angle of arrival, (fl.,=)~ about the '3 actual direction of the main target is (see reference )

for a two-element target.

where

CT~::: ( L Co.s <j) )<. / C£'5 t_jJ R

(h-1)

(4-2)

¢ = angle between the target axis and radar wave front

I(= range to target

)t= ratio of signal amplitude from major and minor

reflectors.

The mean square value of the rate error due to glint

is given by .}"+tP i!

I • )z _ 1 1 ;, S Gr (s)/ ;r( ) c/ t era - .~nr/ · I t- Grcs) I~ s 5 (4-3)

G v -.;-where :.. is the tracking loop transfer function

I .;. r previously defined and s represents the differentiation

performed by the gyro.

*See Appendix III.

4J

For large w., and for t.u,/w~ L. ~ .Z

Gr =- ka. ( S/u;e .,.; ) I 1- Gr sz -1- ka.. {S/t..Vz :1-l)

(4-4)

where,

( 4-5)

Now substituting into the previous expression for the

mean square value of rate error, (see reference 11,

page 458)

( • ) 2 / j_.j...P~ s ~<a- (s/tu4 .,.., )! ~ 2. (.J.}q 2

a: = - ' '-ds 0 -'77,/ -J- sz.,L-.KA.(.:S/tLJz t-1) u.; ~ s'&

(4-6)

Now putting this expression into a form given in a

table of integrals (reference 8, page 369),

( . )'l.. z a. Y"..!..j.ic:¥> s ¥ Ia:, '&.- s ~ o;; = ~ o;; "7z:~"..~)-[S3+t'?+~)s¥<d.+":;-'1)' +u.J;I4}[hf-,>] (4-7)

The solution for the integral given in the table i~

I= - K~ (It- GU2./Wa.,)- I (4-8) wq..£

therefore

( •)z. z z l<a. ( / r UJ;Ict.~z) f- Wz z.. }

o;; = ka C1;; to; z[ .:?KA[t~r ~){/rf:fl)- ~7L4' substituting into the above expression

(4-9)

l.Va / j!/ :1--K. ' .......

"-'G 'l. I/<. JGt_.:.. :z:

f<..= UJ c./ {.Lit:'-

r= t-Uc:. I c»; where t.Uc. is the crossover

frequency of G rl'5)

•

Sample calfulation of seeker rate error ·-·-------Glint-Assumed values for target and kinematics

values are as follows:

I< = e:J, I

t,P = 0

L(_)a_:::: o.oa/ o.d/ ...... ,cl o.oz rad/sec J ., '<-

Substituting into the expression for the aver.::~ge glint

angle,

= using a Ku-Band radar where the wavelength is assumed

to be l.S4cm and the glint noise bandwidth is

uJ; = c.

z 4. ~~a Gt...la-

= ~. S3 ~8.3 Qnd /3~.C rad/5<!!!C. J j

For the above computed values of rTt...' and u.Jt1-) 00 is

plotted as a function of the open tracking loop bandwidth,

UJc in Figure (IV-1). From the curves of figure (IV-1)

for ~c equal to 6.28 rad/sec.

h;)

/.b

\,' P-8 I~

~ ~

~ ~ I( 0., ~ ~ ~ " ~ O.f

az.

4 ~-I

------- ------4.$'

GLINT PRODUCED RATE ERROR AS A FUNCTION OF

TRACKING LOOP BANDWIDTH

FIGURE IV-1

4lc./wz = 3

Gr= & {S/a.~4:J /3,.8 G(./NT NOI .. :UT 8ANbW/L>TH

s~ I (rae~ /sec) OG'.: o. ooa3 /"'UU. I

I I I

I ti.8.3

J I

I I I

I I

I I I

I I I I

I I I I

I I

1 I I I

I I I I

I I / I

I I / /

I I ~.B3 " /

"" / , ., "" / / -/ / --... ........ , .. ---_, .--"' ---- - - --............. ....,. __.,. - ---~..,.. - ----

/.o .r.o /Q.t:> ~'i:?.o /~.o s'oc.o

OPe'"N LOOP TR.ACK/N'G .S;9NDW/.L>T# (;acl'/sec)

+-­'v-)

GLINT BANDWIDTH

(rad/sec)

6.83

68.3

136.6

RATE ERROR

(rad/sec)

0.03

0.12

0.16

44

V. A RECENT GLINT REDUCTION TECHNIQUE

A. General

Recent technological advancements have provided

means for reducing radar tracking errors by utilizing

a principle commonly calJed frequency agility. The

effect of glint on the accuracy of radar angle tracking

has been described in the previous sections of this

report. This section is devoted to describing

analytically, the improvement which can be derived by

using a frequency agile technique.

B. Analytic Improvement Factor

For the purposes of this section, whether or net

glint is a significant problem with a fixed frequency

radar is not considered.

Analysis - The reduction in glint tracking angle noise

afforded by frequency agile radar can be derived on a

very much simplified basis. The basic reason for the

improvement is that the frequency agile radar obtains

more independent samples in a given time than does the

fixed frequency radar.

45

If the frequency agile radar jumps rapidly and far

enough so that each pulse is completely uncorrelated,

the number of independent samples per second is simply

equal to the pulse repetition rate, ~~.

( 5-l)

If the frequency agile radar does not have unlimited

bandwidth, then the number of independent samples per

second is equal to the number of independent frequency

samples that can be obtained in one second. This is

dependent on the total frequency agile bandwidth, the

correlation bandwidth of the target, ~f , and the

correlation time of the glint, -!(t . The correlation

bandwidth of the target is given as

L1t= £_ cD (5-2)

where c is the velocity of light, and D is the length

of the target in the range direction. For example,

a target 15 meters long will have a correlation

bandwidth of 10 MHz. When the number of independent

samples is bandwidth limited, the number of independent

E~Clmples per second is then

~ {,cA) == ~-4)( -i;J (5-3)

46

In a typical case, .B equals 500 MH z, tll' = 101"1#-e , and

-6~ = 0.05 sec. For this case,

~ (.CA) == fs;~") G.:~-)= /Ooo (5-4)

Since we have generally been considering PRF's of over

1,000, we will find that the number of independent

samples is bandwidth limited rather than PRF limited.

The total number of possible independent samples

per second is 1,000; but if we use a random tuning

system, we will have some repetition, and thus some

fraction of the 1,000 will be identical pieces of

information, not independent samples. We will later

show that the average fraction of independent samples

is about 70% to 90% of the maximum possible. Assuming

a value of 80% for the case described (Fi-= O. ) , we

obtain 800 independent samples per second.

Now for a fixed frequency radar, the number of

independent samples per second is simply the reciprocal

of the glint correlation time.

IV ,cp = _!_ per secoNd -t;.

(5-5)

The ratio of the improvement of the frequency agile

radar to the improvement of the fixed frequency is,

(5-6)

As an example, with an agility bandwidth of 500 MHz, a

glint correlation bandwidth of 10 MHz, andn = 0.8

I= )}d. a{ ~~0) = HZ = ~ . .s-

Barton, Reference

pulses integrated

3, gives the effective number of .fr-

as~ where /3n, is the servo

bandwidth. The reduction in glint is proportional to

the square root of the number of pulses integrated;

thus the improvement factor is

(PRF Limited) (5-7)

or

(Bandwidth Limited) (5-8)

whichever is smaller.

For fixed frequency,

.:r,,:r: J = Yt; ..,~N' c 5-9)

Note that for both fixed frequency and frequency agile

radars, the tracking error is reduced by decreasing

the tracking servo bandwidth.

To calculate the fraction of the pulses which are

independent, we will assume that the bandwidth, B , is

broken up into n discrete frequency resolution cells.

'?1... = _!i_ .~ ea..c h.. LJ f. U.J ,·de. L1f

(5-10)

We now generate a group of ~ samples (some are

independent, some may not be). If we plot a typical

histogram, it will look something like Figure V-1.

HISTOGRAM

~ 8 .... -1

-1

~4-1--..f I l Figure V-1

The probability of getting a hit on one trial is~. The """"-

probability of getting a zero in any one resolution cell

on all .e trials (samples) is

Po : (1 - _;._ J :C ( 5-11)

The number of samples is simply the number of pul:=:,e::c·.

in one glint correlation time period: :c equals PR F· t-3-

We will now set out to compute the average number

of empty cells. Let K be the number of empty cells.

Then the average number of empty cells, K , is:

( 5-12)

f'{N.:.PJ = probability of 1<=-D (all have at lea~~t one), or the probability of any one ccJJ not empty = 1- Po

PO<=()J = probability of all cells not empty = (/-~)'>'L

.,P(.k:.J) =probability of 1<=1

?(l<=-1) =probability of one cell empty and all others not empty. The probability of a specific cell being empty and the rest not empty is

( .,..,_, p~.l) = ~ 1- ~)

"1'1 Now there are z ways of choosing which one is empty;

therefore, p (J.<::.!) ::- { >J) ~ {I- ?a) ')11-/

and ) {"Yt.) z. >1-Z.. P{k.=Z ::: z /-;, {1-i~)

The general term is obviously

PfX)t: (~)(~)J<(I- ~)-n-K

This is the binomial distribution. (We could have

possibly arrived at this conclusion by inspection of the

process.) The average number of empty cells {R) if then

~ = 'n ~ = 'n. ( 1- ~) c I"' := -n-)(. Ji- / J) 2-

1 --=n:"' = I - =;;;:_ = I - c I - ::;t_

Substituting M back into equation (6) for the improvement

over a fixed frequency radar, we get:

I = V { 1- c ,_ ~n 1 t; ~ (5-13)

:L = )/( [ A.f.J PA~. L .8 1- 1- ~ , 7fF ( 5-14)

')()

and substituting for Llf

7 ::: )'[I - (I - i!.c;S) Pitt:. I;] .Z .(U3 c

putting in the value of C

.I= Vi/- (1- 3>~JOB)I'Je.,:,(,.;/2b.8 e.oB .I :.13 .x 1a 8

= Vn- ( t- zJI. ,os)-- z. .a &B $.><1a8

(5-15)

(5-16)

(5-18)

Results - The improvement factor is plotted in Figure

(V-2) as a function of the ratio PRF to glint bandwidth.

Note that increasing the agility bandwidth is the moEt

significant factor influencing the improvement factor.

Note that this improvement is with respect to a fixed

frequency radar. Both types can effect a further glint

reduction by averaging the glint over a time period

long compared to the glint correlation time (for

example, by the tracking servo bandwidth). Since this

factor affects both radar types equally, it wasn't

analyzed here.

Analytical Approximations - Some approximations were

used in this analysis, and hence the results can only

be viewed as approximations. For example, equation

(5-15) gives the number of independent samples per

second as the reciprocal of the glint correlation time.

This would only be strictly true if the glint

J(}

5

/0

FREQUENCY AGILITY GLINT IMPROVEMENT FIGURE V-2

L -::: T.4.ttt:~-r t,e-N<; rN- /"?t:rc-R.S

8 - AG/~;;:ry 8/INIJ\~/.OTN- MQ/t!f"~S

,..- .. -------... ---

JJ=£~1~"-- -~--

/ -/ J~M~ ------,; ------------

_.,/ ----, --/, - I / ,..-'

-~ ./-~/ /

/ ..-" // /"'

/ .J' / - - - - - - --- - - - -- 5'" )( 1'0 .3 , .,.. -- ~ -------------/ / ---~--

/1 // // I /,

/ /;/ ;I// II I - - - - - - - - - - - - -- - - - .:J. x Ia 3 I I ..... - -------------.I _.,..- ---------

/ / / / I 1-- - - - - - - - - - - - - - - - - - - - - - - - /X ~0 .3 .... -- -----------------

/ /

50 /00

{p~,e;)/CGttNT SAN.OW.I.OT#)

IS'"

\J1 1\)

autocorrelation function were unit from zero to~~, and

zero elsewhere, which is a physical impos2ibility. This

approximation is shown below.

---FIGURE V-3

GLINT AUTOCORRELATION APPROXHJlATION

For a A further approximation is that i:d- = ~d-10

description of this approximation, see Reference

pages 435 and 436. It is felt that the total uncPrt~inty

introduced by these approximations should be less than

a factor of about 1.5.

VI. CONCLUSIONS

An analytical approach was used to describe the

effect of Glint (angle noise) on the dynamic tracking

characteristics of a missile seeker. The equations

which resulted were too cumbersome for extracting useful

information without the aid of a computer for a target

model with more than two reflecting elements. This

approach does not adequately describe the physical target

because the return energy actually has a statistical

distribution for all but the very simple geometric

bodies.

The statistical approach fashioned after the

classical random walk problem (from statistics) was used

to develop a statistical distribution. The resulting

distribution was Rayleigh and transforming this into

spectral form illustrated that a low frequency

distribution resulted which is similar to present day

glint models derived from experimental data. The results

were then incorporated into the classical statistical

analysis of feedback control systems with the conclusion

that glint is the predominant error source for closing

ranges.

54

In addition, a new technique was discussed and

analysis showed that an im9rovement can be obtained by

reducing the effect of glint through the use of

frequency agile techniq~es.

55

/

VII. APPENDICES

A-PROPORTIONAL NAVIGATION*

l. Definition: A proportional-navigation course is a

/'? /

/ /

course in which the rate of change of

missile heading is directly proportional

to the rate of rotation of the line-of-

sight from the missile to the target.

r = instantaneous missile-to-target range

M = missile position Vr = target velocity VM = missile velocity ¢ = angle between target

rJ#I = r =

J=

velocity vector and line-of-sight missile heading missile heading relative to line-of-sight target position

Figure AI-l, Proportional-Navigation Geometry

As an aid to deriving the radial and transverse

components Figure Al-2 can be used,

~~4 are unit vectors

Figure AI-2, Polar Coordinates

l *See reference

-Since the vector r is v- units long in the C'r-

direction,

(Al-l)

and the velocity is given by

(Al-2)

where

• . The unit vectors er and e<fJ must be determined. To

do this we moye from point P to point 9 in Figure Al-l, e9-t-~ -,. ,.~e,.

FIGURE AI-3

Evaluation of Polar Coordinate Unit Vectors

now in the limit

· de,.. cJ <J = fJ e ¢> e.,. = cl c- ~ . . eq9 :: de~> d~.: - t? e;.

elf: cit:: the velocity vector may be written from Al-2-3-4

as

(Al-?,)

(Al-4)

( Al- 5)

The velocity for the geometry in Figure Al-l is

given as • •

V= r- -rrcf) (A1-6)

where t"' is the velocity component taken in direction of •

~ and r~ is the component taken in the direction normal

to r .

(Al-?)

and using the sign convention adopted in Figure Al-3 •

(AJ-f\)

and

(Al-9)

where (Al-9) represents the proportional mentioned in

the definition where the constant a is called the

navigation constant or the navigational correction.

Now (Al-9) can be integrated directly and the result i~

c/1'1 = <?. ¢f + ~o (A 1 -1 0 )

From reference 1 if Q =1 and <id=(J, a pure pur::-:ui t

course results, and if q=/ and sic is fixed then a •

deviated pursuit course results. If¢1 .=. o , we have a

constant-bearing course. This was the reason for

selecting this guidance mode as it is representative of

various guidance schemes.

The equations of motion A7-8-9 cannot be solved

in closed form except when Q = Z.. The solution for

q : a can be found in reference 1 along with additional

discussions of the dynamic characteristics of the

proportional navigation scheme.

!(_) )7

B-FAR FIELD CRITERION FOR

p

Figure AII-1, Antenna Pattern Geometry

A general criterion for the incident plane wav0

cr:Lterion can be found in Chapter 6, of reference r).

The development is based on the principle:3 of optic:;.

From Figure AII-1, an examination of the geometry

reveals that for large ~ and a finite receiving

aperature the phase variation across the aperature fc

zero, i.e., a plane wave.

For point P , a wave emitted from the source~ will

have a phase E3hift (~'')!<. and for the same wave arriving

at A the phase shift will be ()_77) S. Ho'tJevcr, S i ~:

dependent on the range ~ and the aperature width D ,

since

Therefore the phase difference between P and A- due to

a wave traveling from ~ is,

or

t ,()

changing from rectangular coordinates (x,y) to polar

coordinates (E,G), the distribution becomes

u.; { c-- B)= 1::-- e- e/nrro J JT?'ID(, (3-JS)

Now if this expression is integrated with respect to G, z

cv { €) = :-J E e - e /n oo no;; (3-39)

and since ().= x "+~ 2. , we have ( o-::: t.= z..J

~ (_ tr)d a-: d rr - o-/"YLCT;, ( 3-LjO) -:>'LODe

This is the Rayleigh distribution, where the

average cross section is a-::-n cro. Therefore, the

average echoing area of the assembly is the sum of the

echoing areas of the individual elements. Now, two

glint models are available for continued analysis.

From Barton (reference 3 ) the power density for a

standard glint model is,

fCUJ) = i(o} w~Y. LU~ ... +l.U ~

where Wg is the half-power frequency (noise bandwidth) {3)

2 Wo.. L ;t

and is the zero frequency spectral density o~

average amplitude squared.

lUa_ = rate of change of aspect angle

L = maximum dimension of the target measured

normal to the direction of the radar betlrn

and to the axis of rotation

~ = radar wave length

A second expression can be derived by translating

the developed model into spectrum form. To calculate

the power spectral density,

p (w):: f-;. -it.u r/?C 'rd r --where, Rcr =A e- Aid r (Rayleigh Power Distribution)

and, A:: 'l'n 00

Then .,p

Prw) = A/_ e-./""'~- Ardr

= A L-(co:s c:.v r-~·s;-n!U/ r )e -A~ r = A;_-Cos 41 Y' e- A t;y r - A.//-~$ OU-4..1 re- ...,. Y"d r

Returning to equation (3-42), and assuming that

RC/ represents the autocorrelation function (which

by definition is an even function).

Then I? ( T) = A e- A,-.

and ./tw) =

From the previous development, and the model

(3-43)

described one observes that the glint spectrum is of

the low-pass type, and from Barton, reference 3 , the

bandwidth is proportional to the spread in radial

velocities within the target, divided by the radar

wavelength (i.e., W:J= 277/'?c.Uq.L)). The general glint ,A.

spectral density is demonstrated in figure III-5.

35

B-FAR FIELD CRITERION FOR

Figure AII-1, Antenna Pattern Geometry

A general criterion for the incident plane wave

criterion can be found in Chapter 6, of reference 9.

The development is based on the principles of optics.

From Figure AII-1, an examination of the geometry

reveals that for large ~ and a finite receiving

aperature the phase variation across the aperature is

zero, i.e., a plane wave.

For point P , a wave emitted from the source Cj- will

have a phase shift(~'')!? and for the same wave arriving

at A the phase shift will be ()_11) S . However, S i.s

dependent on the range ~ and the aperature width D ,

l. ~ since S=- [ Rzl- J)/-f] z.

Therefore the phase difference between P and 4 due to

a wave traveling from 'r is,

or

60

now assume

D;z = I<R._,

then

and

~ 2llj,.tl? { ( 1 ,LJ<z) 1._ I}= A~

expanding the radical using the binomial theorem and

disregarding higher order terms,

- RK-z. LJ¢ = ~. A z_.

now LJ(i = ?jl{l+jl<~I)R

L:J()-= ~Tf (~;R) now an accepted minimum phase shift is

).. /u.. then K Zj< .4. /1/f" i!

and K 1. 6 -'A/tt.R. =- }.f SR

and from the previous assumption,

or

The problem which exists when considering a plane

incident wave for glint analysis is that the target

completely fills the seeker beamwidth and for a

closing seeker the incident wave will not be a plane

wave. For a purely analytic analysis this factor would

have to be incorporated into the analysis, and this

technique used in the analysis of a two and three

element target. The statistical approach could be

handled by employing a two dimensional statistical

model, however, the phase distortion in the incident

wave along the target (aperture) will be incorporated

as part of the random phase distribution returning

from the target. Therefore, a constant-phase front

is assumed for the incident wave at the target.

62

C-ANGULAR ERROR FOR THE TWO-ELEMENT TARGET

For the purposes of completing the analysis an

expression will be developed for the angular error

of a two-element target. The development will be

similar to the approach used in reference 3 pages 85

through 89.

For the purposes of the development refer to

figure AIII-1.

,-------._. A ...... ----, '--- -1- I

,;" ------\ ~ I _....,.-,....---t::------ ,,:' • • '-:.....- .._ -- \

' - -":::::::1;. -..- - t!!/!!1 ..... ---?.... I I

e = s?.tJ/A.'T a"f"'~ ~..;::

/~~ ... d'<!.>ca L a.n .,te;,na,

eA'e.-n,ht"

............ '....... / / - - - - - -· ,tJ

Figure AIII-l,-Radar Angle Error Coordinates (One Coordinate)

For a simultaneous lobing radar, the angle error

is derived by comparing the difference in the two

received channels (for one coordinate) with the

composite energy received (or the sum of the total

return energy received). The voltage derived from

each lobe is passed through an amplifier, rtetected,

and then the difference is used to obtain the error

by dividing the difference by the sum. The sum channel

providing a means for calibrating the error channels

for varying target positions relative to the antenna

line-of-sight.

Now again referring to Figure AIII-1, the boresight

axis is directed at the center of the target AB , and

for a small angular displacement, the slope of the

antenna lobes can be considered constant, and will be

denoted by cg.. .

E"A and E"i3 represent the received voltages from the

target whene~o , and their phase difference is given

by rP. The total RF voltage received by the upper lobe is,

E"'"" = E4 {I+) t9) +-'=-1.3 (/-; t§)e ./f# (A3-1)

and for the lower lobe,

~::;-;_ = En ( 1 -; <¢) .;- t::'/3 ( 1 f-) 6J.) e J~ (A3-2)

the difference voltage is

£D = X (/E.-a)~ /L:L..J / (AJ-3)

and substituting A3-12 into A3-3 and simplifying

£ D = J<. "'/;; & [ ~4 -z..- e 8 *"] (A 3·-·4)

where A = amplifier gain, the sum channel is

E_,= K(IE~/~ l~!z) which simplifies to,

~-s ~ ;2,K..[ ~ ~ E,a z.l- .;JI=~tb c c:s ?1 J (A3-6)

the error voltage as previously defined is the ratio

of the difference and the sum signals or, Et:> _ -?K 7 (6)) [ E;q ~ e-a ']

E.s o7k.[ e.4 z.., c-~ z. -i-2li;q.Esct:Joc).:J

now for a single target,

(AJ-7)

E"'.t> , z. /'J £ -z. -::- =- ~I< t!r;:T A = 29B (AJ-8) e$ .2~f?A2-

Therefore the apparent shift (error) at the boreshight

angle for a two-element would be

or

E'.i> l?"s

E.o' = €;:,

:J.;; G {. 6:4 z._ £/3 t)/( E~ t.r E13 z. r 2EAE8 CJo:s t;l) &>;B

' substituting 6>

I for e".i>;e5 and {;J = eo I e s

e~';;; e[ 1- {_e>~IEA)z} / + {u-"leA- t+- 2€8/t:A- C.<'~P

(A~3-9)

Now, a relationship between the angle off-set ~

and the relative phase relationship f is required.

Figure AIII-2, Wavefront Geometry

From figure AIII-2,

R -z. .:: .R- .-12/z s .uv ~ }<; - R + _a/~ s .1 n tf'

Now for a reflecting target, the total phase

difference is given by,

LJ~ = -5'7T { .!?, -~) ft.

and

Therefore

and,

For small angles, sinO~ t9 (in radians)

therefore* 6' .::: _f C! & ..s t/)

,;2.~

t;:) I-- _£ C? ,d .5 # /- k. "L

,;2 ,I( I f- )::. z. ~ ;:z. ,K C' t!J .s 'f'

(AJ-10)

(AJ-11)

(AJ-12)

(AJ-13)

where J< is the ratio of echoE; from target A and 13 , and

_/ is the separation of the target elements, and cf is

given by

*For the purposes of this discussion the target major axis was assumed to be nearly perpendicular to the radar line-of-sight.

For the purposes of this analysis assume

.1< z ~..( .z &"~ (.Lt:!d.s?J )jt~-R(' 1 +:J.kc~s ¢)

:!- R ~~.s t11 I ') R /~ ( / r.:2J< C'~ :s ¢/ (A3-l4)

VIII. BIBLIOGRAPHY

(1) Locke, ArthurS. Guidance, D.Van Nostrand Co.

Inc. 1955

(2) Povejsil, Raven, Waterman, Airborne Radar,

D. Van Nostrand Co. Inc., 1961

(3) Barton, David K., Modern Radar Analysis, Prentice­

Hall Inc., 1964

(4) Jordan, Edward C., Electromagnetic Waves and

Radiating Systems, Prentice-Hall Inc., 1968

(5) Shinners, Stanley M., Control System Design,

John Wiley & Sons, Inc., 1964

(6) Beckmann, Petr, Probability in Communication

Engineering, Harcourt, Brace & World, Inc., 1967

(7) Pipes, Louis A., Applied Mathematics for Engineers

and Physicists, McGraw-Hill, Inc., 1958

(8) James, Nichols, and Phillips, Theory of Servo­

mechanisms, McGraw-Hill, 1947

(9) Silver, Samuel, Microwave Antenna Theory and

Design, McGraw-Hill, 1947

(10) Schwartz, Mischa, Information Transmission,

Modulation, and Noise, McGraw-Hill, 1959

(11) Truxal, John G., Automatic Feedback Control System

Synthesis, McGraw-Hill, 1955

68

IX. VITA

Franklin Delano Hockett was born on April 17, 1933,

in Des Moines, Iowa. He received his primary and

secondary education in Des Moines, and Runnells, Iowa.

He attended the State University of Iowa, Iowa City,

Iowa, and received his Bachelor of Science Degree in

Electrical Engineering during June, 1959.

While the author was employed at Collins Radio Co.,

Cedar Rapids, Iowa, he took graduate courses from

Iowa State University, Ames, Iowa during the years

1962 - 64. He has been enrolled in the St. Louis

Graduate Extension Center of the University of Missouri,

at Rolla, since September 1964. During this time

interval he has been employed at McDonnell-Douglas

Corporation and more recently at Conductron-Missouri.

69